Continuity and strict monotonicity conjecture for the tilted critical parameter

Let $G$ be a connected, locally finite graph, and let $\Gamma\subseteq\operatorname{Aut}(G)$ be quasi-transitive and nonunimodular. For each $\lambda$, let $p_c(\lambda)$ denote the corresponding tilted critical parameter. Continuity and strict monotonicity conjecture. The function

$$\lambda\mapsto p_c(\lambda)$$

is continuous and strictly increasing on $(-\infty,1/2]$.

The source states that this is unproved, while proving continuity and strict increase on the subset where $p_c(\lambda)<p_t$. The conjecture concerns the full interval $(-\infty,1/2]$.